∫(dx/√(a² + x²)) = ln|(x + √(a² + x²))/a| + c

∫(dx/√(a² + x²)) = ln|(x + √(a² + x²))/a| + c

∫(dx/√(a² + x²)) = ln|(x + √(a² + x²))/a| + c

Answer

  1. Explanation:
    • Step 1: Recall the integral formula
      • The integral (\int\frac{dx}{\sqrt{a^{2}+x^{2}}}) is a well - known integral in calculus. The antiderivative of the function (\frac{1}{\sqrt{a^{2}+x^{2}}}) with respect to (x) is (\ln|x + \sqrt{a^{2}+x^{2}}|+C), where (C) is the constant of integration. This formula can be derived using trigonometric substitution. Let (x = a\tan\theta), then (dx=a\sec^{2}\theta d\theta) and (\sqrt{a^{2}+x^{2}}=\sqrt{a^{2}+a^{2}\tan^{2}\theta}=a\sec\theta).
      • (\int\frac{dx}{\sqrt{a^{2}+x^{2}}}=\int\frac{a\sec^{2}\theta d\theta}{a\sec\theta}=\int\sec\theta d\theta).
      • The integral of (\sec\theta) is (\ln|\sec\theta+\tan\theta| + C). Substituting back (\tan\theta=\frac{x}{a}) and (\sec\theta=\frac{\sqrt{a^{2}+x^{2}}}{a}), we get (\ln|x+\sqrt{a^{2}+x^{2}}|+C).
  2. Answer:
    • The given equation (\int\frac{dx}{\sqrt{a^{2}+x^{2}}}=\ln\left|x+\sqrt{a^{2}+x^{2}}\right|+C) is a correct integral formula. So the statement is True.